Uncertainty Quantification (UQ):
Since my PhD years, I have worked extensively on uncertainty quantification, with a focus on developing efficient numerical methods. My PhD study led to the development of generalized polynomial chaos (gPC) method.
A concise one-semester textbook on stochastic methods related to UQ was published by Princeton University Press in 2010.
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Approximation Theory:
Multivariate approximation theory and algorithms related to (orthogonal) polynomials, Gaussian Process (GP).
Efficient sampling strategies and Design of Experiments (DoE)
Machine Learning for Scientific Computing:
Data driven modeling of dynamical systems. In particular, the flow map learning (FML) methods.
DNN modeling of Partial differential equations (PDEs)
Deep learning of stochastic differential equations (SDE)
Digital Twins
Multi University Research Initiative (MURI), 2024 - 2029, "Mathematics for Digital Twins".
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